The matrix M is defined asand has eigenvalues 5and andminus;2. The matrix Qis formed as Q = M3- 4M2- 2MWhich of the following is/are the eigenvalue(s) of matrix Q?a)15b)25c)-20d)-30Correct answer is option 'A,C'. Can you explain this answer?

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The matrix M is defined as

and has eigenvalues 5 and −2. The matrix Q is formed as Q = M³ - 4M² - 2M
Which of the following is/are the eigenvalue(s) of matrix Q?

a)
15
b)
25
c)
-20
d)
-30

Correct answer is option 'A,C'. Can you explain this answer?

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The matrix M is defined asand has eigenvalues 5and −2. The matri...

Properties of eigenvalues-

Eigenvalues of unitary and orthogonal matrices are of unit modulus |λ| = 1

If λ1, λ2…….λn are the eigenvalues of A, then kλ1, kλ2…….kλn are eigenvalues of kA
If λ1, λ2…….λn are the eigenvalues of A, then 1/λ1, 1/λ2…….1/λn are eigenvalues of A-1
If λ1, λ2…….λn are the eigenvalues of A, then λ1k, λ2k…….λnk are eigenvalues of Ak

Eigenvalues of A = Eigen Values of AT (Transpose)

Sum of Eigen Values = Trace of A (Sum of diagonal elements of A)

Product of Eigen Values = |A|

Maximum number of distinct eigenvalues of A = Size of A

Calculation:

The given eigenvalues are 5 and −2.

Q = M3 - 4M2 - 2M

Eigen values of M³ are 125 and -8,4M² are 100 and 16, 2M are 10 and -4.

SO eigen values of Q, λ1= 125-100-10 = 15,λ2 = -8-16-(-4)=-20

So the eigen values are 15 and -20

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The matrix M is defined asand has eigenvalues 5and −2. The matrix Qis formed as Q = M3- 4M2- 2MWhich of the following is/are the eigenvalue(s) of matrix Q?a)15b)25c)-20d)-30Correct answer is option 'A,C'. Can you explain this answer?

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